1 (* Title: HOL/Tools/meson.ML
3 Author: Lawrence C Paulson, Cambridge University Computer Laboratory
4 Copyright 1992 University of Cambridge
6 The MESON resolution proof procedure for HOL.
8 When making clauses, avoids using the rewriter -- instead uses RS recursively
10 NEED TO SORT LITERALS BY # OF VARS, USING ==>I/E. ELIMINATES NEED FOR
11 FUNCTION nodups -- if done to goal clauses too!
14 signature BASIC_MESON =
16 val size_of_subgoals : thm -> int
17 val make_cnf : thm list -> thm -> thm list
18 val finish_cnf : thm list -> thm list
19 val make_nnf : thm -> thm
20 val make_nnf1 : thm -> thm
21 val skolemize : thm -> thm
22 val make_clauses : thm list -> thm list
23 val make_horns : thm list -> thm list
24 val best_prolog_tac : (thm -> int) -> thm list -> tactic
25 val depth_prolog_tac : thm list -> tactic
26 val gocls : thm list -> thm list
27 val skolemize_prems_tac : thm list -> int -> tactic
28 val MESON : (thm list -> tactic) -> int -> tactic
29 val best_meson_tac : (thm -> int) -> int -> tactic
30 val safe_best_meson_tac : int -> tactic
31 val depth_meson_tac : int -> tactic
32 val prolog_step_tac' : thm list -> int -> tactic
33 val iter_deepen_prolog_tac : thm list -> tactic
34 val iter_deepen_meson_tac : thm list -> int -> tactic
35 val meson_tac : int -> tactic
36 val negate_head : thm -> thm
37 val select_literal : int -> thm -> thm
38 val skolemize_tac : int -> tactic
39 val make_clauses_tac : int -> tactic
46 val not_conjD = thm "meson_not_conjD";
47 val not_disjD = thm "meson_not_disjD";
48 val not_notD = thm "meson_not_notD";
49 val not_allD = thm "meson_not_allD";
50 val not_exD = thm "meson_not_exD";
51 val imp_to_disjD = thm "meson_imp_to_disjD";
52 val not_impD = thm "meson_not_impD";
53 val iff_to_disjD = thm "meson_iff_to_disjD";
54 val not_iffD = thm "meson_not_iffD";
55 val conj_exD1 = thm "meson_conj_exD1";
56 val conj_exD2 = thm "meson_conj_exD2";
57 val disj_exD = thm "meson_disj_exD";
58 val disj_exD1 = thm "meson_disj_exD1";
59 val disj_exD2 = thm "meson_disj_exD2";
60 val disj_assoc = thm "meson_disj_assoc";
61 val disj_comm = thm "meson_disj_comm";
62 val disj_FalseD1 = thm "meson_disj_FalseD1";
63 val disj_FalseD2 = thm "meson_disj_FalseD2";
65 val depth_limit = ref 2000;
67 (**** Operators for forward proof ****)
70 (** First-order Resolution **)
72 fun typ_pair_of (ix, (sort,ty)) = (TVar (ix,sort), ty);
73 fun term_pair_of (ix, (ty,t)) = (Var (ix,ty), t);
75 val Envir.Envir {asol = tenv0, iTs = tyenv0, ...} = Envir.empty 0
77 (*FIXME: currently does not "rename variables apart"*)
78 fun first_order_resolve thA thB =
79 let val thy = theory_of_thm thA
80 val tmA = concl_of thA
81 fun match pat = Pattern.first_order_match thy (pat,tmA) (tyenv0,tenv0)
82 val Const("==>",_) $ tmB $ _ = prop_of thB
83 val (tyenv,tenv) = match tmB
84 val ct_pairs = map (pairself (cterm_of thy) o term_pair_of) (Vartab.dest tenv)
85 in thA RS (cterm_instantiate ct_pairs thB) end
86 handle _ => raise THM ("first_order_resolve", 0, [thA,thB]);
88 (*raises exception if no rules apply -- unlike RL*)
89 fun tryres (th, rls) =
90 let fun tryall [] = raise THM("tryres", 0, th::rls)
91 | tryall (rl::rls) = (th RS rl handle THM _ => tryall rls)
94 (*Permits forward proof from rules that discharge assumptions. The supplied proof state st,
95 e.g. from conj_forward, should have the form
96 "[| P' ==> ?P; Q' ==> ?Q |] ==> ?P & ?Q"
97 and the effect should be to instantiate ?P and ?Q with normalized versions of P' and Q'.*)
98 fun forward_res nf st =
99 let fun forward_tacf [prem] = rtac (nf prem) 1
100 | forward_tacf prems =
101 error ("Bad proof state in forward_res, please inform lcp@cl.cam.ac.uk:\n" ^
104 cat_lines (map string_of_thm prems))
106 case Seq.pull (ALLGOALS (METAHYPS forward_tacf) st)
108 | NONE => raise THM("forward_res", 0, [st])
111 (*Are any of the logical connectives in "bs" present in the term?*)
113 let fun has (Const(a,_)) = false
114 | has (Const("Trueprop",_) $ p) = has p
115 | has (Const("Not",_) $ p) = has p
116 | has (Const("op |",_) $ p $ q) = member (op =) bs "op |" orelse has p orelse has q
117 | has (Const("op &",_) $ p $ q) = member (op =) bs "op &" orelse has p orelse has q
118 | has (Const("All",_) $ Abs(_,_,p)) = member (op =) bs "All" orelse has p
119 | has (Const("Ex",_) $ Abs(_,_,p)) = member (op =) bs "Ex" orelse has p
124 (**** Clause handling ****)
126 fun literals (Const("Trueprop",_) $ P) = literals P
127 | literals (Const("op |",_) $ P $ Q) = literals P @ literals Q
128 | literals (Const("Not",_) $ P) = [(false,P)]
129 | literals P = [(true,P)];
131 (*number of literals in a term*)
132 val nliterals = length o literals;
135 (*** Tautology Checking ***)
137 fun signed_lits_aux (Const ("op |", _) $ P $ Q) (poslits, neglits) =
138 signed_lits_aux Q (signed_lits_aux P (poslits, neglits))
139 | signed_lits_aux (Const("Not",_) $ P) (poslits, neglits) = (poslits, P::neglits)
140 | signed_lits_aux P (poslits, neglits) = (P::poslits, neglits);
142 fun signed_lits th = signed_lits_aux (HOLogic.dest_Trueprop (concl_of th)) ([],[]);
144 (*Literals like X=X are tautologous*)
145 fun taut_poslit (Const("op =",_) $ t $ u) = t aconv u
146 | taut_poslit (Const("True",_)) = true
147 | taut_poslit _ = false;
150 let val (poslits,neglits) = signed_lits th
151 in exists taut_poslit poslits
153 exists (member (op aconv) neglits) (HOLogic.false_const :: poslits)
155 handle TERM _ => false; (*probably dest_Trueprop on a weird theorem*)
158 (*** To remove trivial negated equality literals from clauses ***)
160 (*They are typically functional reflexivity axioms and are the converses of
161 injectivity equivalences*)
163 val not_refl_disj_D = thm"meson_not_refl_disj_D";
165 (*Is either term a Var that does not properly occur in the other term?*)
166 fun eliminable (t as Var _, u) = t aconv u orelse not (Logic.occs(t,u))
167 | eliminable (u, t as Var _) = t aconv u orelse not (Logic.occs(t,u))
168 | eliminable _ = false;
170 fun refl_clause_aux 0 th = th
171 | refl_clause_aux n th =
172 case HOLogic.dest_Trueprop (concl_of th) of
173 (Const ("op |", _) $ (Const ("op |", _) $ _ $ _) $ _) =>
174 refl_clause_aux n (th RS disj_assoc) (*isolate an atom as first disjunct*)
175 | (Const ("op |", _) $ (Const("Not",_) $ (Const("op =",_) $ t $ u)) $ _) =>
177 then refl_clause_aux (n-1) (th RS not_refl_disj_D) (*Var inequation: delete*)
178 else refl_clause_aux (n-1) (th RS disj_comm) (*not between Vars: ignore*)
179 | (Const ("op |", _) $ _ $ _) => refl_clause_aux n (th RS disj_comm)
180 | _ => (*not a disjunction*) th;
182 fun notequal_lits_count (Const ("op |", _) $ P $ Q) =
183 notequal_lits_count P + notequal_lits_count Q
184 | notequal_lits_count (Const("Not",_) $ (Const("op =",_) $ _ $ _)) = 1
185 | notequal_lits_count _ = 0;
187 (*Simplify a clause by applying reflexivity to its negated equality literals*)
189 let val neqs = notequal_lits_count (HOLogic.dest_Trueprop (concl_of th))
190 in zero_var_indexes (refl_clause_aux neqs th) end
191 handle TERM _ => th; (*probably dest_Trueprop on a weird theorem*)
194 (*** The basic CNF transformation ***)
196 (*Estimate the number of clauses in order to detect infeasible theorems*)
197 fun nclauses (Const("Trueprop",_) $ t) = nclauses t
198 | nclauses (Const("op &",_) $ t $ u) = nclauses t + nclauses u
199 | nclauses (Const("Ex", _) $ Abs (_,_,t)) = nclauses t
200 | nclauses (Const("All",_) $ Abs (_,_,t)) = nclauses t
201 | nclauses (Const("op |",_) $ t $ u) = nclauses t * nclauses u
202 | nclauses _ = 1; (* literal *)
204 (*Replaces universally quantified variables by FREE variables -- because
205 assumptions may not contain scheme variables. Later, call "generalize". *)
207 let val newname = gensym "mes_"
208 val spec' = read_instantiate [("x", newname)] spec
211 (*Used with METAHYPS below. There is one assumption, which gets bound to prem
212 and then normalized via function nf. The normal form is given to resolve_tac,
213 presumably to instantiate a Boolean variable.*)
214 fun resop nf [prem] = resolve_tac (nf prem) 1;
216 (*Any need to extend this list with
217 "HOL.type_class","Code_Generator.eq_class","ProtoPure.term"?*)
219 exists_Const (fn (c,_) => c mem_string ["==", "==>", "all", "prop"]);
221 fun apply_skolem_ths (th, rls) =
222 let fun tryall [] = raise THM("apply_skolem_ths", 0, th::rls)
223 | tryall (rl::rls) = (first_order_resolve th rl handle THM _ => tryall rls)
226 (*Conjunctive normal form, adding clauses from th in front of ths (for foldr).
227 Strips universal quantifiers and breaks up conjunctions.
228 Eliminates existential quantifiers using skoths: Skolemization theorems.*)
229 fun cnf skoths (th,ths) =
230 let fun cnf_aux (th,ths) =
231 if not (HOLogic.is_Trueprop (prop_of th)) then ths (*meta-level: ignore*)
232 else if not (has_conns ["All","Ex","op &"] (prop_of th))
233 then th::ths (*no work to do, terminate*)
234 else case head_of (HOLogic.dest_Trueprop (concl_of th)) of
235 Const ("op &", _) => (*conjunction*)
236 cnf_aux (th RS conjunct1, cnf_aux (th RS conjunct2, ths))
237 | Const ("All", _) => (*universal quantifier*)
238 cnf_aux (freeze_spec th, ths)
240 (*existential quantifier: Insert Skolem functions*)
241 cnf_aux (apply_skolem_ths (th,skoths), ths)
242 | Const ("op |", _) => (*disjunction*)
244 (METAHYPS (resop cnf_nil) 1) THEN
245 (fn st' => st' |> METAHYPS (resop cnf_nil) 1)
246 in Seq.list_of (tac (th RS disj_forward)) @ ths end
247 | _ => (*no work to do*) th::ths
248 and cnf_nil th = cnf_aux (th,[])
250 if nclauses (concl_of th) > 20
251 then (Output.debug ("cnf is ignoring: " ^ string_of_thm th); ths)
252 else cnf_aux (th,ths)
255 (*Convert all suitable free variables to schematic variables,
256 but don't discharge assumptions.*)
257 fun generalize th = Thm.varifyT (forall_elim_vars 0 (forall_intr_frees th));
259 fun make_cnf skoths th = cnf skoths (th, []);
261 (*Generalization, removal of redundant equalities, removal of tautologies.*)
262 fun finish_cnf ths = filter (not o is_taut) (map (refl_clause o generalize) ths);
265 (**** Removal of duplicate literals ****)
267 (*Forward proof, passing extra assumptions as theorems to the tactic*)
268 fun forward_res2 nf hyps st =
271 (METAHYPS (fn major::minors => rtac (nf (minors@hyps) major) 1) 1)
274 | NONE => raise THM("forward_res2", 0, [st]);
276 (*Remove duplicates in P|Q by assuming ~P in Q
277 rls (initially []) accumulates assumptions of the form P==>False*)
278 fun nodups_aux rls th = nodups_aux rls (th RS disj_assoc)
279 handle THM _ => tryres(th,rls)
280 handle THM _ => tryres(forward_res2 nodups_aux rls (th RS disj_forward2),
281 [disj_FalseD1, disj_FalseD2, asm_rl])
284 (*Remove duplicate literals, if there are any*)
286 if has_duplicates (op =) (literals (prop_of th))
287 then nodups_aux [] th
291 (**** Generation of contrapositives ****)
293 (*Associate disjuctions to right -- make leftmost disjunct a LITERAL*)
294 fun assoc_right th = assoc_right (th RS disj_assoc)
297 (*Must check for negative literal first!*)
298 val clause_rules = [disj_assoc, make_neg_rule, make_pos_rule];
300 (*For ordinary resolution. *)
301 val resolution_clause_rules = [disj_assoc, make_neg_rule', make_pos_rule'];
303 (*Create a goal or support clause, conclusing False*)
304 fun make_goal th = (*Must check for negative literal first!*)
305 make_goal (tryres(th, clause_rules))
306 handle THM _ => tryres(th, [make_neg_goal, make_pos_goal]);
308 (*Sort clauses by number of literals*)
309 fun fewerlits(th1,th2) = nliterals(prop_of th1) < nliterals(prop_of th2);
311 fun sort_clauses ths = sort (make_ord fewerlits) ths;
313 (*True if the given type contains bool anywhere*)
314 fun has_bool (Type("bool",_)) = true
315 | has_bool (Type(_, Ts)) = exists has_bool Ts
316 | has_bool _ = false;
318 (*Is the string the name of a connective? Really only | and Not can remain,
319 since this code expects to be called on a clause form.*)
320 val is_conn = member (op =)
321 ["Trueprop", "op &", "op |", "op -->", "Not",
322 "All", "Ex", "Ball", "Bex"];
324 (*True if the term contains a function--not a logical connective--where the type
325 of any argument contains bool.*)
326 val has_bool_arg_const =
328 (fn (c,T) => not(is_conn c) andalso exists (has_bool) (binder_types T));
330 (*Raises an exception if any Vars in the theorem mention type bool; they
331 could cause make_horn to loop! Also rejects functions whose arguments are
332 Booleans or other functions.*)
334 not (exists (has_bool o fastype_of) (term_vars t) orelse
335 not (Term.is_first_order ["all","All","Ex"] t) orelse
336 has_bool_arg_const t orelse
339 (*Create a meta-level Horn clause*)
340 fun make_horn crules th = make_horn crules (tryres(th,crules))
343 (*Generate Horn clauses for all contrapositives of a clause. The input, th,
344 is a HOL disjunction.*)
345 fun add_contras crules (th,hcs) =
346 let fun rots (0,th) = hcs
347 | rots (k,th) = zero_var_indexes (make_horn crules th) ::
348 rots(k-1, assoc_right (th RS disj_comm))
349 in case nliterals(prop_of th) of
351 | n => rots(n, assoc_right th)
354 (*Use "theorem naming" to label the clauses*)
355 fun name_thms label =
356 let fun name1 (th, (k,ths)) =
357 (k-1, Thm.name_thm (label ^ string_of_int k, th) :: ths)
359 in fn ths => #2 (foldr name1 (length ths, []) ths) end;
361 (*Is the given disjunction an all-negative support clause?*)
362 fun is_negative th = forall (not o #1) (literals (prop_of th));
364 val neg_clauses = List.filter is_negative;
367 (***** MESON PROOF PROCEDURE *****)
369 fun rhyps (Const("==>",_) $ (Const("Trueprop",_) $ A) $ phi,
370 As) = rhyps(phi, A::As)
371 | rhyps (_, As) = As;
373 (** Detecting repeated assumptions in a subgoal **)
375 (*The stringtree detects repeated assumptions.*)
376 fun ins_term (net,t) = Net.insert_term (op aconv) (t,t) net;
378 (*detects repetitions in a list of terms*)
379 fun has_reps [] = false
380 | has_reps [_] = false
381 | has_reps [t,u] = (t aconv u)
382 | has_reps ts = (Library.foldl ins_term (Net.empty, ts); false)
383 handle Net.INSERT => true;
385 (*Like TRYALL eq_assume_tac, but avoids expensive THEN calls*)
386 fun TRYING_eq_assume_tac 0 st = Seq.single st
387 | TRYING_eq_assume_tac i st =
388 TRYING_eq_assume_tac (i-1) (eq_assumption i st)
389 handle THM _ => TRYING_eq_assume_tac (i-1) st;
391 fun TRYALL_eq_assume_tac st = TRYING_eq_assume_tac (nprems_of st) st;
393 (*Loop checking: FAIL if trying to prove the same thing twice
394 -- if *ANY* subgoal has repeated literals*)
396 if exists (fn prem => has_reps (rhyps(prem,[]))) (prems_of st)
397 then Seq.empty else Seq.single st;
400 (* net_resolve_tac actually made it slower... *)
401 fun prolog_step_tac horns i =
402 (assume_tac i APPEND resolve_tac horns i) THEN check_tac THEN
403 TRYALL_eq_assume_tac;
405 (*Sums the sizes of the subgoals, ignoring hypotheses (ancestors)*)
406 fun addconcl(prem,sz) = size_of_term(Logic.strip_assums_concl prem) + sz
408 fun size_of_subgoals st = foldr addconcl 0 (prems_of st);
411 (*Negation Normal Form*)
412 val nnf_rls = [imp_to_disjD, iff_to_disjD, not_conjD, not_disjD,
413 not_impD, not_iffD, not_allD, not_exD, not_notD];
415 fun make_nnf1 th = make_nnf1 (tryres(th, nnf_rls))
417 forward_res make_nnf1
418 (tryres(th, [conj_forward,disj_forward,all_forward,ex_forward]))
421 (*The simplification removes defined quantifiers and occurrences of True and False.
422 nnf_ss also includes the one-point simprocs,
423 which are needed to avoid the various one-point theorems from generating junk clauses.*)
425 [simp_implies_def, Ex1_def, Ball_def, Bex_def, if_True,
426 if_False, if_cancel, if_eq_cancel, cases_simp];
427 val nnf_extra_simps =
428 thms"split_ifs" @ ex_simps @ all_simps @ simp_thms;
431 HOL_basic_ss addsimps nnf_extra_simps
432 addsimprocs [defALL_regroup,defEX_regroup,neq_simproc,let_simproc];
434 fun make_nnf th = case prems_of th of
435 [] => th |> rewrite_rule (map safe_mk_meta_eq nnf_simps)
436 |> simplify nnf_ss (*But this doesn't simplify premises...*)
438 | _ => raise THM ("make_nnf: premises in argument", 0, [th]);
440 (*Pull existential quantifiers to front. This accomplishes Skolemization for
441 clauses that arise from a subgoal.*)
443 if not (has_conns ["Ex"] (prop_of th)) then th
444 else (skolemize (tryres(th, [choice, conj_exD1, conj_exD2,
445 disj_exD, disj_exD1, disj_exD2])))
447 skolemize (forward_res skolemize
448 (tryres (th, [conj_forward, disj_forward, all_forward])))
449 handle THM _ => forward_res skolemize (th RS ex_forward);
452 (*Make clauses from a list of theorems, previously Skolemized and put into nnf.
453 The resulting clauses are HOL disjunctions.*)
454 fun make_clauses ths =
455 (sort_clauses (map (generalize o nodups) (foldr (cnf[]) [] ths)));
457 (*Convert a list of clauses (disjunctions) to Horn clauses (contrapositives)*)
460 (distinct Drule.eq_thm_prop (foldr (add_contras clause_rules) [] ths));
462 (*Could simply use nprems_of, which would count remaining subgoals -- no
463 discrimination as to their size! With BEST_FIRST, fails for problem 41.*)
465 fun best_prolog_tac sizef horns =
466 BEST_FIRST (has_fewer_prems 1, sizef) (prolog_step_tac horns 1);
468 fun depth_prolog_tac horns =
469 DEPTH_FIRST (has_fewer_prems 1) (prolog_step_tac horns 1);
471 (*Return all negative clauses, as possible goal clauses*)
472 fun gocls cls = name_thms "Goal#" (map make_goal (neg_clauses cls));
474 fun skolemize_prems_tac prems =
475 cut_facts_tac (map (skolemize o make_nnf) prems) THEN'
478 (*Expand all definitions (presumably of Skolem functions) in a proof state.*)
479 fun expand_defs_tac st =
480 let val defs = filter (can dest_equals) (#hyps (crep_thm st))
481 in PRIMITIVE (LocalDefs.def_export false defs) st end;
483 (*Basis of all meson-tactics. Supplies cltac with clauses: HOL disjunctions*)
484 fun MESON cltac i st =
486 (EVERY [rtac ccontr 1,
488 EVERY1 [skolemize_prems_tac negs,
489 METAHYPS (cltac o make_clauses)]) 1,
490 expand_defs_tac]) i st
491 handle THM _ => no_tac st; (*probably from make_meta_clause, not first-order*)
493 (** Best-first search versions **)
495 (*ths is a list of additional clauses (HOL disjunctions) to use.*)
496 fun best_meson_tac sizef =
498 THEN_BEST_FIRST (resolve_tac (gocls cls) 1)
499 (has_fewer_prems 1, sizef)
500 (prolog_step_tac (make_horns cls) 1));
502 (*First, breaks the goal into independent units*)
503 val safe_best_meson_tac =
504 SELECT_GOAL (TRY Safe_tac THEN
505 TRYALL (best_meson_tac size_of_subgoals));
507 (** Depth-first search version **)
509 val depth_meson_tac =
510 MESON (fn cls => EVERY [resolve_tac (gocls cls) 1,
511 depth_prolog_tac (make_horns cls)]);
514 (** Iterative deepening version **)
516 (*This version does only one inference per call;
517 having only one eq_assume_tac speeds it up!*)
518 fun prolog_step_tac' horns =
519 let val (horn0s, hornps) = (*0 subgoals vs 1 or more*)
520 take_prefix Thm.no_prems horns
521 val nrtac = net_resolve_tac horns
522 in fn i => eq_assume_tac i ORELSE
523 match_tac horn0s i ORELSE (*no backtracking if unit MATCHES*)
524 ((assume_tac i APPEND nrtac i) THEN check_tac)
527 fun iter_deepen_prolog_tac horns =
528 ITER_DEEPEN (has_fewer_prems 1) (prolog_step_tac' horns);
530 fun iter_deepen_meson_tac ths =
532 case (gocls (cls@ths)) of
533 [] => no_tac (*no goal clauses*)
535 (THEN_ITER_DEEPEN (resolve_tac goes 1)
537 (prolog_step_tac' (make_horns (cls@ths)))));
539 fun meson_claset_tac ths cs =
540 SELECT_GOAL (TRY (safe_tac cs) THEN TRYALL (iter_deepen_meson_tac ths));
542 val meson_tac = CLASET' (meson_claset_tac []);
545 (**** Code to support ordinary resolution, rather than Model Elimination ****)
547 (*Convert a list of clauses (disjunctions) to meta-level clauses (==>),
548 with no contrapositives, for ordinary resolution.*)
550 (*Rules to convert the head literal into a negated assumption. If the head
551 literal is already negated, then using notEfalse instead of notEfalse'
552 prevents a double negation.*)
553 val notEfalse = read_instantiate [("R","False")] notE;
554 val notEfalse' = rotate_prems 1 notEfalse;
556 fun negated_asm_of_head th =
557 th RS notEfalse handle THM _ => th RS notEfalse';
559 (*Converting one clause*)
560 fun make_meta_clause th =
561 if is_fol_term (prop_of th)
562 then negated_asm_of_head (make_horn resolution_clause_rules th)
563 else raise THM("make_meta_clause", 0, [th]);
565 fun make_meta_clauses ths =
567 (distinct Drule.eq_thm_prop (map make_meta_clause ths));
569 (*Permute a rule's premises to move the i-th premise to the last position.*)
571 let val n = nprems_of th
572 in if 1 <= i andalso i <= n
573 then Thm.permute_prems (i-1) 1 th
574 else raise THM("select_literal", i, [th])
577 (*Maps a rule that ends "... ==> P ==> False" to "... ==> ~P" while suppressing
579 val negate_head = rewrite_rule [atomize_not, not_not RS eq_reflection];
581 (*Maps the clause [P1,...Pn]==>False to [P1,...,P(i-1),P(i+1),...Pn] ==> ~P*)
582 fun select_literal i cl = negate_head (make_last i cl);
585 (*Top-level Skolemization. Allows part of the conversion to clauses to be
586 expressed as a tactic (or Isar method). Each assumption of the selected
587 goal is converted to NNF and then its existential quantifiers are pulled
588 to the front. Finally, all existential quantifiers are eliminated,
589 leaving !!-quantified variables. Perhaps Safe_tac should follow, but it
590 might generate many subgoals.*)
592 fun skolemize_tac i st =
593 let val ts = Logic.strip_assums_hyp (List.nth (prems_of st, i-1))
596 (fn hyps => (cut_facts_tac (map (skolemize o make_nnf) hyps) 1
597 THEN REPEAT (etac exE 1))),
598 REPEAT_DETERM_N (length ts) o (etac thin_rl)] i st
600 handle Subscript => Seq.empty;
602 (*Top-level conversion to meta-level clauses. Each clause has
603 leading !!-bound universal variables, to express generality. To get
604 disjunctions instead of meta-clauses, remove "make_meta_clauses" below.*)
605 val make_clauses_tac =
608 let val ts = Logic.strip_assums_hyp prop
613 (map forall_intr_vars
614 (make_meta_clauses (make_clauses hyps))) 1)),
615 REPEAT_DETERM_N (length ts) o (etac thin_rl)]
619 (*** setup the special skoklemization methods ***)
621 (*No CHANGED_PROP here, since these always appear in the preamble*)
623 val skolemize_meth = Method.SIMPLE_METHOD' HEADGOAL skolemize_tac;
625 val make_clauses_meth = Method.SIMPLE_METHOD' HEADGOAL make_clauses_tac;
627 val skolemize_setup =
629 [("skolemize", Method.no_args skolemize_meth,
630 "Skolemization into existential quantifiers"),
631 ("make_clauses", Method.no_args make_clauses_meth,
632 "Conversion to !!-quantified meta-level clauses")];
636 structure BasicMeson: BASIC_MESON = Meson;